IB MYP 3 Mathematics 1.3 Irrational Numbers, Square Roots and Cube Roots Study Notes - New Syllabus
IB MYP 3 Mathematics 1.3 Irrational Numbers, Square Roots and Cube Roots Study Notes
IB MYP 3 Mathematics 1.3 Irrational Numbers, Square Roots and Cube Roots Study Notes at IITian Academy focus on specific topics and types of questions asked in the actual exam. Study Notes focus on the IB MYP 3 Mathematics syllabus with guiding questions of
Rational number: Can be written as \(\dfrac{p}{q}\), where \(p,q\) are integers and \(q\neq0\).
Irrational number: Cannot be written as a fraction of two integers; its decimal is non-terminating and non-recurring.
Perfect square: A number whose square root is an integer.
Square root: A number that produces the original number when multiplied by itself.
Perfect cube: A number whose cube root is an integer.
Cube root: A number that produces the original number when multiplied by itself three times.
Key distinction: \(\sqrt{a}\) is the principal non-negative square root, while \(x^2=a\) may have two solutions.
1.3 – Irrational Numbers, Square Roots and Cube Roots
Numbers can be classified into different groups based on how they can be represented. In this topic, we focus on irrational numbers, square roots and cube roots. These ideas help us work with numbers that cannot always be written as simple fractions.
Rational and Irrational Numbers
A rational number can be written in the form:
\(\dfrac{p}{q}\), where \(p\) and \(q\) are integers and \(q\neq0\).
An irrational number cannot be written as a fraction of two integers. Its decimal representation is non-terminating and non-recurring.
| Type | Decimal Representation | Examples |
|---|---|---|
| Rational | Terminates or repeats | \(\dfrac{3}{4}=0.75\), \(\dfrac{1}{3}=0.333\ldots\) |
| Irrational | Never terminates and never repeats | \(\sqrt{2}\), \(\sqrt{5}\), \(\pi\) |
A decimal that continues forever is not automatically irrational.
For example, \(0.333\ldots\) is rational because the digits repeat and \(0.333\ldots=\dfrac{1}{3}\).
An irrational decimal continues forever without a repeating pattern.
√ Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number.
\(\sqrt{a}=b\) means \(b^2=a\).
For example:
\(\sqrt{25}=5\), because \(5^2=25\).
The numbers \(1,4,9,16,25,36,\ldots\) are called perfect squares because their square roots are integers.
| Number | Square Root |
|---|---|
| \(1\) | \(\sqrt{1}=1\) |
| \(4\) | \(\sqrt{4}=2\) |
| \(9\) | \(\sqrt{9}=3\) |
| \(16\) | \(\sqrt{16}=4\) |
| \(25\) | \(\sqrt{25}=5\) |
| \(36\) | \(\sqrt{36}=6\) |
Principal Square Root
The symbol \(\sqrt{}\) represents the principal, non-negative square root.
\(\sqrt{49}=7\)
However, if we solve the equation \(x^2=49\), there are two solutions:
\(x=7\) or \(x=-7\)
\(\sqrt{49}=7\), but solving \(x^2=49\) gives \(x=\pm7\).
Square Roots of Non-Perfect Squares
Not every positive integer is a perfect square. For example, \(2,3,5,6,7,8,\ldots\) are not perfect squares.
Therefore, their square roots are irrational numbers.
- \(\sqrt{2}\approx1.41421356\ldots\)
- \(\sqrt{3}\approx1.7320508\ldots\)
- \(\sqrt{5}\approx2.2360679\ldots\)
These decimal values continue forever without repeating.
Estimating Square Roots
A square root can be estimated by identifying the two consecutive perfect squares between which the number lies.
For example, to estimate \(\sqrt{30}\):
\(25<30<36\)
Therefore:
\(5<\sqrt{30}<6\)
Since \(30\) is closer to \(25\) than to \(36\), \(\sqrt{30}\) is closer to \(5\) than to \(6\).
Using a calculator:
\(\sqrt{30}\approx5.477\)
1. Find the perfect square below the number.
2. Find the perfect square above the number.
3. Take the square roots of those perfect squares.
4. Place the required square root between them.
∛ Cube Roots
The cube root of a number is a value that, when multiplied by itself three times, gives the original number.
\(\sqrt[3]{a}=b\) means \(b^3=a\).
For example:
\(\sqrt[3]{27}=3\), because \(3^3=27\).
| Perfect Cube | Cube Root |
|---|---|
| \(1\) | \(\sqrt[3]{1}=1\) |
| \(8\) | \(\sqrt[3]{8}=2\) |
| \(27\) | \(\sqrt[3]{27}=3\) |
| \(64\) | \(\sqrt[3]{64}=4\) |
| \(125\) | \(\sqrt[3]{125}=5\) |
| \(216\) | \(\sqrt[3]{216}=6\) |
Cube Roots of Negative Numbers
Unlike square roots, cube roots can be taken of negative numbers because a negative number multiplied by itself three times remains negative.
\((-3)^3=-27\)
Therefore:
\(\sqrt[3]{-27}=-3\)
Similarly:
\(\sqrt[3]{-64}=-4\)
Square root: \(\sqrt{-9}\) is not a real number.
Cube root: \(\sqrt[3]{-27}=-3\) is a real number.
Simplifying Square Roots
Some square roots can be simplified by taking perfect-square factors outside the square root.
For example:
\(\sqrt{72}=\sqrt{36\times2}\)
\(=6\sqrt{2}\)
The factor \(36\) was chosen because it is the largest perfect-square factor of \(72\).
Another example:
\(\sqrt{48}=\sqrt{16\times3}=4\sqrt{3}\)
For \(a\geq0\) and \(b\geq0\):
\(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\)
Comparing Irrational Numbers
Irrational numbers can be compared by estimating their decimal values or by comparing their squares when the numbers are positive.
For example, compare \(\sqrt{7}\) and \(\sqrt{10}\).
Since:
\(7<10\)
and both numbers are positive:
\(\sqrt{7}<\sqrt{10}\)
Approximately:
- \(\sqrt{7}\approx2.646\)
- \(\sqrt{10}\approx3.162\)
When working with irrational numbers:
1. Identify whether the number is rational or irrational.
2. Look for perfect squares or perfect cubes.
3. Simplify roots where possible.
4. Estimate using nearby perfect squares or cubes when necessary.
5. Use a calculator when a decimal approximation is required.
Example 1:
Consider the numbers \(\sqrt{18}\), \(0.625\), \(\sqrt{25}\) and \(0.121212\ldots\).
a) Identify which numbers are rational and which are irrational.
b) Simplify \(\sqrt{18}\).
c) Estimate \(\sqrt{18}\) to the nearest tenth.
▶️ Answer/Explanation
Answer
a) Classifying the numbers
\(0.625\) is rational because it terminates:
\(0.625=\dfrac{5}{8}\)
\(\sqrt{25}\) is rational because:
\(\sqrt{25}=5\)
\(0.121212\ldots\) is rational because the digits repeat.
\(\sqrt{18}\) is irrational because \(18\) is not a perfect square.
Therefore:
Rational: \(0.625,\;\sqrt{25},\;0.121212\ldots\)
Irrational: \(\sqrt{18}\)
b) Simplify \(\sqrt{18}\)
\(\sqrt{18}=\sqrt{9\times2}\)
\(=3\sqrt{2}\)
c) Estimate
Since:
\(16<18<25\)
we know:
\(4<\sqrt{18}<5\)
Using a calculator:
\(\sqrt{18}\approx4.243\)
Therefore, to the nearest tenth:
\(\sqrt{18}\approx4.2\)
Example 2:
A rectangular prism has a volume of \(216\text{ cm}^3\).
a) Find the cube root of \(216\).
b) Simplify \(\sqrt{72}\).
c) Explain why \(\sqrt{72}\) is irrational.
d) Estimate \(\sqrt{72}\) to the nearest whole number.
▶️ Answer/Explanation
Answer
a) Cube root
We need the number that gives \(216\) when cubed.
\(6^3=6\times6\times6=216\)
Therefore:
\(\sqrt[3]{216}=6\)
b) Simplify \(\sqrt{72}\)
Use the largest perfect-square factor of \(72\):
\(\sqrt{72}=\sqrt{36\times2}\)
\(=6\sqrt{2}\)
c) Why is \(\sqrt{72}\) irrational?
Although \(72\) contains a perfect-square factor, \(72\) itself is not a perfect square. Therefore, \(\sqrt{72}\) cannot be written as a fraction of two integers.
Since:
\(8^2=64\) and \(9^2=81\)
\(72\) lies between two consecutive perfect squares, so \(\sqrt{72}\) is irrational.
d) Estimate \(\sqrt{72}\)
\(64<72<81\)
Therefore:
\(8<\sqrt{72}<9\)
Using a calculator:
\(\sqrt{72}\approx8.485\)
To the nearest whole number:
\(\sqrt{72}\approx8\)